Problem: Marquise has $200$ meters of fencing to build a rectangular garden. The garden's area (in square meters) as a function of the garden's width $x$ (in meters) is modeled by: $A(x)=-x^2+100x$ What side width will produce the maximum garden area?
Answer: The garden's area is modeled by a quadratic function, whose graph is a parabola. The maximum area is reached at the vertex. So in order to find when that happens, we need to find the vertex's $w$ -coordinate. The vertex's $x$ -coordinate is the average of the two zeros, so let's find those first. $\begin{aligned} A(x)&=0 \\\\ -x^2+100x&=0 \\\\ x^2-100x&=0 \\\\ x(x-100)&=0 \\\\ \swarrow &\searrow \\\\ x=0\text{ or }&x-100=0 \\\\ x={0}\text{ or }&x={100} \end{aligned}$ Now let's take the zeros' average: $\dfrac{({0})+({100})}{2}=\dfrac{100}{2}=50$ In conclusion, the maximum garden area occurs when the width is $50$ meters.